Distributional data analysis via quantile functions and its application to modeling digital biomarkers of gait in Alzheimer’s Disease

Abstract

Summary With the advent of continuous health monitoring with wearable devices, users now generate their unique streams of continuous data such as minute-level step counts or heartbeats. Summarizing these streams via scalar summaries often ignores the distributional nature of wearable data and almost unavoidably leads to the loss of critical information. We propose to capture the distributional nature of wearable data via user-specific quantile functions (QF) and use these QFs as predictors in scalar-on-quantile-function-regression (SOQFR). As an alternative approach, we also propose to represent QFs via user-specific L-moments, robust rank-based analogs of traditional moments, and use L-moments as predictors in SOQFR (SOQFR-L). These two approaches provide two mutually consistent interpretations: in terms of quantile levels by SOQFR and in terms of L-moments by SOQFR-L. We also demonstrate how to deal with multi-modal distributional data via Joint and Individual Variation Explained using L-moments. The proposed methods are illustrated in a study of association of digital gait biomarkers with cognitive function in Alzheimers disease. Our analysis shows that the proposed methods demonstrate higher predictive performance and attain much stronger associations with clinical cognitive scales compared to simple distributional summaries.

Scalar on time-by-distribution regression and its application for modelling associations between daily-living physical activity and cognitive functions in Alzheimer’s Disease

“…(1) www.nature.com/scientificreports/ Distributional data analysis using subject-specific quantile functions. Distributional data analysis can capture and model distributional aspect of wearable observations via subject-specific probability density functions (pdf), cumulative distribution functions (CDF), or quantile functions 21 . If we ignore the temporal information by suppressing the time index t, we can denote by X ik , k = 1, .…”

“…A reduced capacity of physical activity can be observed for the AD samples compared to CNC across upper quantile levels such as p > 0.75 . Following the approach of Ghosal et al 21 , the subject-specific quantile functions of PA can be used for modelling Y i using scalar-on-function regression (SOFR) (3) adjusted for Z i . SOFR model is as follows where the functional regression coefficient β(p) captures the distributional effect of the PA quantile function Q i (p) on the response of interest Y i .…”

“…In the case β(p) = β , a constant, one again get back the generalized linear model (1) from model (3), since Xi = 1 0 Q i (p)dp. Ghosal et al 21 re-represented SOFR model for quantile function predictors via L-moments 33 . L-moments are defined as the expectation of a linear combination of order statistics.…”

“…The main advantages of L-moments is the existence of all moments, if first moment exist, their uniqueness and robustness. For SOQFR Ghosal et al 21 adapted an alternative representation of L-moments as projections of quantile functions on Legendre polynomial basis, given by Here P r (p) is the shifted Legendre polynomial (LP) of degree r defined as The shifted Legendre polynomials form an orthogonal basis of L 2 [0, 1] . Using the LP decomposition for subjectspecific quantile functions…”

Processing of Accelerometry Data with GGIR in Motor Activity Research Consortium for Health

“…as functional observations recorded over 24 hr (John et al, 2019;Karas et al, 2022;Leroux et al, 2019;Migueles, Cadenas-Sanchez, et al, 2019;Ramsay, 2006;Wrobel et al, 2021;Xiao et al, 2015). Distributional information can be captured via distributional data analysis that encodes information via subject-specific distributional representations such as quantile functions, cumulative distribution functions, probability density functions, and others (Ghosal et al, 2021(Ghosal et al, , 2022. Time-series aspects of the data such as fragmentation can be accounted via various measures capturing the duration in and the frequency of transitioning between active/sedentary behaviors (Di et al, 2017;Paraschiv-Ionescu et al, 2013).…”